V. Give the correct translation of the underlined part.

1. Up to the XIX century axioms were considered as basic self-evident truth   about the concepts involved.

a) считались                b) считаются                       с) считают

2. Mathematical theorems must be deductively established and proved.

a) должны будут дедуктивно основываться и доказываться

b) должны быть дедуктивно основаны и доказаны

c) должны были быть дедуктивно основаны и доказаны

3. Much of the scientific knowledge is proved by deductive reasoning.

a) доказывает                                       b) доказывается

c) будет доказано

4. New theorems are proved constantly, even in such old subjects as algebra and geometry and the current developments are as important as the older results.

a) старые результаты                                        b) самые старые результаты

c) более старые результаты

5. In ancient times the axioms and theorems were regarded as necessary.

a) необходимо                                 b) необходимые

c) были необходимы

Text 13.

The “Pythagorean” theorem is one of the most important propositions in the entire realm of geometry. There is no doubt, how­ever, that the “Pythagorean property”: c² =a² + b² was known prior to the time of Pythagoras; there existed clay tablet texts which con­tained columns of figures related to Pythagorean triples. The frequent text­book reference to Egyptian “rope-stretchers” and their knotted surveying ropes as proof that these ancients knew the theorem  was  erro­neous. While it is known that the Egyptians realized as early as 2000 B. C. that 4²+ 3²=5², there is no evidence that the Egyptians knew or were able to prove the right angle property of the figure involved. Pythagoras is credited with the proof of this property which is true for all right triangles, and  for all natural numbers. Although much of this information was known to the ancients of earlier times, the deductive aspect of geometry was exploited and advanced considerably in the work of the Pythagoreans.

The mysticism of this celebrated school aroused the suspicion and dislike of the people who finally drove the Pythagoreans out Crotona. A Greek seaport in Southern Italy and burnt their buildings. Pythagoras was murdered but his followers were scattened to other Greek centres and continued his teachings. The Pythagoreans were credited with giving the subject of mathematics special and independent status. They were the first group to treat mathematical concepts as abstractions and they distinguished mathematical theory from practices or calculations. They proved the fundamental theorems of plane and solid geometry and of  “arithmetica”- the theory of numbers.

More widely known than the Pythagoreans was the Academy of Plato which had Aristotle as its most distinguished students. The latter then founded his own school at the time of Plato’s death pupils were the most famous philosophers, mathematicians and astronomers of their age. Under Plato’s influence they emphasized pure mathematics to the extent of ignoring all practical applications and they added immensely to the range of mathematics.

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